Hypercomplex Mathematical Morphology

The natural ordering of grey levels is used in classical mathematical morphology for scalar images to define the erosion/dilation and the evolved operators. Various operators can be sequentially applied to the resulting images always using the same ordering. In this paper we propose to consider the result of a prior transformation to define the imaginary part of a complex image, where the real part is the initial image. Then, total orderings between complex numbers allow defining subsequent morphological operations between complex pixels. More precisely, the total orderings are lexicographic cascades with the local modulus and phase values of these complex images. In this case, the operators take into account simultaneously the information of the initial image and the processed image. In addition, the approach can be generalized to the hypercomplex representation (i.e., real quaternion) by associating to each image three different operations, for instance directional filters. Total orderings initially introduced for colour quaternions are used to define the evolved morphological transformations. Effects of these new operators are illustrated with different examples of filtering.